The Research Of Two Classes Of The Lyapunov System With Three-Order Nilpotent Singular Points

This thesis is devoted to quasi-Lyapunov constants, center focus and limit cycle bifurcation of two classes of the nilpotent singular points in the planar polynomial differential system. It is composed of four chapters.In chapter1, the historical background and the present progress of problems about center conditions and limit cycle bifurcation of the Lyapunov system are introduced and summarized. Meanwhile,the main work of this paper is simply concluded.In chapter2, the successor functions and focal value of three-order nilpotent singular point in generalized polar coordination are given, while three-order nilpotent singular points are divided into three categories,then came to the the corresponding center integration and inverse integrating factor.According to inverse integrating factor recursive formula are derived, then base on the computer system-Mathematica, calculate the quasi-Lyapunov constants.In chapter3, according to the recursive formula quasi-Lyapunov constants and computer system-Mathematica, eight quasi-Lyapunov constants of three-order nilpotent singular points in a class of quartic differential system can be calculated, and the necessary and sufficient condition when origin becomes a central focus are discussed, that is to say such quartic system can branch in8limit cycles.In chapter4, add a few quintic polynomial in the quartic System, at the premise of the different parameters to zero, twelve quasi-Lyapunov constants of three-order nilpotent singular points in a class of quintic differential system are got. Obtaining the corresponding system of analytic first integral, the necessary and sufficient condition when origin becomes a central focus are derived, then proves that such quartic system can branch in12limit cycles.